Evaluating surface data

ABSTRACT

Systems, methods, and instructions encoded in a computer-readable medium can perform operations related to evaluating surface data. Geodetic data for a plurality of surface locations are received. The geodetic data may include surface gradient information and/or surface elevation information for multiple surface locations. A set of constraining relationships is generated based on the geodetic data. The set of constraining relationships relates undetermined values of surface elevation movement and/or undetermined values of surface gradient movement to measured surface elevation changes and/or measured surface gradient changes. Some or all of the constraining relationships include multiple undetermined values. Particular values for surface elevation movements and/or particular values for surface gradient movements are calculated for multiple surface locations based on determining a solution to the set of constraining relationships. In some implementations, a minimum curvature surface may be generated deterministically based on the geodetic data and the particular values identified using the constraining relationships.

This Application is a Divisional Application of and claims the benefitof priority to U.S. patent application Ser. No. 12/956,609, filed onNov. 30, 2010 and entitled “Evaluating Surface Data”, the contents ofwhich are hereby incorporated by reference.

BACKGROUND

The present disclosure relates to evaluating surface data.

Geodetic measurements provide information about the Earth's surface. Forexample, Global Positioning System (GPS) measurements can providethree-dimensional coordinates (e.g., longitude, latitude, and elevation)for locations on the Earth's surface, Interferometric Synthetic ApertureRadar (InSAR) can provide change in position data for locations on theEarth's surface, and tiltmeter measurements can provide tilt data (e.g.,indicating change in elevation gradient) for locations on the Earth'ssurface. In some instances, geodetic measurements over time can be usedto detect temporal changes in the Earth's surface. Analyzing temporalchanges in the Earth's surface may provide information aboutsubterranean structures, resources, and events occurring beneath theEarth's surface.

SUMMARY

This specification describes technologies relating to evaluating surfacedata.

In a general aspect, geodetic data are used to identify temporal changesin surface elevations and/or surface gradients. In some instances, theidentified changes in surface elevation and/or the identified changes insurface gradient may be used with the geodetic data to generate asurface deformation model of a geographic region.

In some aspects, methods, systems, apparatus, and computer programsencoded on computer storage devices are configured to perform operationsfor evaluating surface data. Geodetic data for a plurality of surfacelocations are received. The geodetic data may include surface gradientinformation and/or surface elevation information for multiple surfacelocations. A set of constraining relationships is generated based on thegeodetic data. The set of constraining relationships relatesundetermined values of surface elevation movement and/or undeterminedvalues of surface gradient movement to measured surface elevationchanges and/or measured surface gradient changes. Some or all of theconstraining relationships include multiple undetermined values.Particular values for surface elevation movements and/or particularvalues for surface gradient movements are calculated for multiplesurface locations based on determining a solution to the set ofconstraining relationships. In some implementations, a minimum curvaturesurface may be generated deterministically based on the geodetic dataand the particular values identified using the constrainingrelationships.

Implementations may include one or more of the following features. Thegeodetic data include surface elevation information for a first subsetof the surface locations and surface gradient information for a secondsubset of the surface locations. The set of constraining relationshipsrelates undetermined values for temporal changes in surface gradients atthe first subset of surface locations and undetermined values fortemporal changes in surface elevations at the second subset of surfacelocations to the surface elevation information and the surface gradientinformation included in the geodetic data. Identifying particular valuesfor temporal changes in surface elevation at each surface location inthe subset includes identifying particular values for temporal changesin surface gradient at each of the first subset of locations andparticular values for temporal changes in surface elevation at each ofthe second subset of surface locations. The geodetic data includesurface gradient information and surface elevation information for athird subset of the surface locations, and the set of constrainingrelationships includes the surface gradient information and surfaceelevation information for the third subset of the surface locations.Each of multiple undetermined values are included in multiple differentconstraining relationships. Each of the constraining relationships thatincludes multiple undetermined values, taken by itself, constrainswithout determining the undetermined values in the relationship.

Additionally or alternatively, implementations may include one or moreof the following features. The set of constraining relationshipsincludes a system of linear relationships. Identifying the particularsurface gradient values (i.e., the particular values for the temporalchanges in surface gradient) for the first subset of locations and theparticular surface elevation values (i.e., the particular values for thetemporal changes in surface elevation) for the second subset oflocations includes solving the system of linear relationships for theparticular surface gradient values and the particular surface elevationvalues. Generating the set of constraining relationships includesgenerating one or more matrices. Solving the set of constrainingrelationships includes inverting one or more of the matrices.Identifying the particular surface elevation values and the particularsurface gradient values includes solving the set of constrainingrelationships based on Gaussian elimination or Gauss-Jordan elimination.

Additionally or alternatively, implementations may include one or moreof the following features. The geodetic data include surface coordinateinformation for each of the surface locations. Neighboring pairs of thesurface locations are identified based on the surface coordinates. Theneighboring pairs of surface locations are identified based ongenerating a Delaunay triangulation of the surface locations. TheDelaunay triangulation includes Delaunay connecting lines between eachof the neighboring pairs of surface locations. Each of the constrainingrelationships is based on the geodetic data for a neighboring pair ofthe surface locations.

Additionally or alternatively, implementations may include one or moreof the following features. A constraining relationship constrains thevalues of the changes in surface gradients at a neighboring pair ofsurface locations to values that result in a minimum surface curvaturebetween the neighboring pair of locations. A constraining relationshipconstrains the values of the changes in surface elevations at aneighboring pair of surface locations to a lowest order change inelevation. The distance between a pair of surface locations may berepresented as l based on the surface coordinates of the surfacelocations. For each neighboring pair of surface locations where thegeodetic data include a value t₁ for a temporal change in surfacegradient at a first point in the neighboring pair and a value t₂ for atemporal change in surface gradient at a second point in the neighboringpair, the set of constraining relationships constrains an undeterminedvalue h₁ for a temporal change in surface elevation at the first pointand an undetermined value h₂ for a temporal change in surface elevationat the second point by a relationship of the form h₂−h₁=½(t₁+t₂)l. Foreach neighboring pair where the geodetic data include a value h1 for atemporal change in surface elevation at a first point in the neighboringpair and a value t₂ for a temporal change in surface gradient at asecond point in the neighboring pair, the set of constrainingrelationships constrains an undetermined value t₁ for a temporal changein surface gradient at the first point and an undetermined value h₂ fora temporal change in surface elevation at the second point by arelationship of the form 2h₂−t₁l=2h₁+t₂l. For each neighboring pairwhere the geodetic data include a value h₁ for a temporal change insurface elevation at a first point in the neighboring pair and a valueh₂ for a temporal change in surface elevation at a second point in theneighboring pair, the set of constraining relationships constrains anundetermined value t₁ for a temporal change in surface gradient at thefirst point and an undetermined value t₂ for a temporal change insurface gradient at the second point by a relationship of the form

${t_{1} + t_{2}} = {\frac{2}{l}{\left( {h_{1} - h_{2}} \right).}}$

Additionally or alternatively, implementations may include one or moreof the following features. Parameters of elevation curves between theneighboring pairs of surface locations are determined based on thereceived geodetic data, the particular surface gradient values (i.e.,the particular values for the temporal changes in surface gradient atthe first subset of surface locations), and the particular surfaceelevation values (i.e., the particular values for the temporal changesin surface elevation at the second subset of surface locations). Eachelevation curve represents a surface deformation between a neighboringpair of surface locations. The surface locations correspond to a regionon the Earth's surface, and the method further includes calculatingtemporal changes in elevation for other surface locations in the regionbased on the parameters of one or more of the elevation curves.Determining the parameters of the elevation curves includes generatingparticular values of coefficients for terms of polynomial elevationcurves. The polynomial elevation curves can include third orderpolynomials. The particular values of the coefficients for the terms ofthe polynomial elevation curves correspond to a minimum curvaturesurface. The particular values of the coefficients correspond to aunique minimum curvature surface for the particular surface gradientvalues, the particular surface elevation values, and measured elevationand gradient values in the geodetic data. A graphical representation ofgeographical surface deformation can be generated based on thepolynomial elevation curves. The geographical surface deformation canrepresent a change in the shape of the geographic surface over aspecified time period. The geographical surface deformation can becorrelated with field activities associated with the geographic regionand the time period.

Additionally or alternatively, implementations may include one or moreof the following features. The geodetic data may be stored in adatabase, and one or more of the operations may be performed by dataprocessing apparatus. A measurement subsystem may acquire the geodeticdata for multiple measurement locations that correspond to the surfacelocations represented in the geodetic data. The measurement subsystemincludes a tiltmeter array that generates the surface gradientinformation. The measurement subsystem includes a GPS and/or an InSARsystem that generates the surface elevation information. One or more ofthe surface locations corresponds to a measurement location of atiltmeter station. One or more of the surface locations corresponds to ameasurement location of a GPS receiver and/or a measurement location ofan InSAR system. The surface gradient measurements may include tiltmeterdata from a tiltmeter array in an area in a geographic region, and thesurface elevation measurements may include InSAR data for another areaoutside a perimeter of the tiltmeter array.

Additionally or alternatively, implementations may include one or moreof the following features. The geodetic data include geodetic data for aplurality of time periods. A set of constraining relationships isgenerated for each of the plurality of time periods. The constrainingrelationship for each time period is solved to identify temporal changesin surface elevation and/or temporal changes in surface gradient for thetime period. Models of geographic surface deformation are generated foreach time period.

One or more implementations of the subject matter described in thisspecification can be implemented so as to realize one or more of thefollowing advantages. A surface model may be generated in less time andin a deterministic manner. For example, one or more of the techniquesdescribed herein may be used to generate a minimum curvature surfacefaster than would be generated using some conventional techniques thatrely on stochastic algorithms. Increased calculation speed may allowcalculations to be performed at a finer time scale, so that solutionscan be presented closer to real time and with greater temporal detail.As such, increased calculation speed may allow more precise correlationof surface deformation to field activities. A surface model may begenerated based on geodetic data that includes only surface elevationinformation or only surface gradient information for multiple surfacelocations. For example, one or more of the disclosed techniques may beused to generate a minimum curvature surface based on geodetic data thatincludes InSAR data for surface locations outside of a geographic regioncovered by a tiltmeter array. Integration of tiltmeter data with InSARdata outside the tiltmeter array may allow more comprehensive analysisof surface deformation beyond comparing the tiltmeter solution with theInSAR-derived measurements.

The details of one or more embodiments of the subject matter describedin this specification are set forth in the accompanying drawings and thedescription below. Other features, aspects, and advantages of thesubject matter will become apparent from the description, the drawings,and the claims.

DESCRIPTION OF DRAWINGS

FIG. 1A is a schematic diagram showing an example surface evaluationsystem.

FIG. 1B is a schematic diagram showing example geodetic data points andconnecting lines generated by the surface evaluation system 100 shown inFIG. 1A.

FIG. 2 is a schematic diagram showing example geodetic data points andconnecting lines.

FIG. 3 is a flow chart showing an example process for evaluating surfacedata.

FIGS. 4A and 4B are diagrams showing example surface deformation models.

FIG. 5 is a schematic diagram showing aspects of an example computingsystem.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

FIG. 1A is a schematic diagram showing an example surface evaluationsystem 100. The example surface evaluation system 100 in FIG. 1Aincludes a measurement subsystem 101 and a computing subsystem 103. Insome implementations, a surface evaluation system may include additionaland/or different features, components and/or subsystems. At a highlevel, the measurement subsystem 101 performs measurements to acquireinformation about a geographic surface 111 in a geographic region 102,and the information acquired by the measurement subsystem 101 isprocessed by the computing subsystem 103 to evaluate, analyze, and/ormodel aspects of the geographic region 102. In some instances, thecomponents and subsystems of the surface evaluation system 100 mayperform additional and/or different types of functions.

In some implementations, the computing subsystem 103 can calculate thesmoothest possible surface from measurements of surface elevation,measurements of surface gradient, and/or measurements of both surfaceelevation and surface gradient at each surface location. Someconventional techniques require elevation and gradient information atevery surface location. Such conventional techniques typically do notincorporate InSAR data from surface regions substantially outside thetiltmeter array. Such conventional techniques utilize stochasticalgorithms that can be time consuming and indeterminate in thatsubsequent iterations can produce different results. The computingsubsystem 103 may deterministically generate a minimum curvature surfacebased on InSAR data, tiltmeter data, and GPS data for multiple surfacelocations in any combination, which may include data points having onlysurface elevation data, only surface gradient data, or both elevationand gradient data. The computing subsystem 103 may generate a solutionfaster than some conventional systems. In some instances, an infinitenumber of different surfaces could theoretically be fit to a set ofsurface elevation and surface gradient measurements, and it may bedesirable to generate a fit having a substantially minimum curvature.For example, it may be desirable to generate the surface that exhibitsthe lowest curvature and corresponds to the least complicated surfacethat satisfies the measurements. In some instances, the curvature of asurface can be estimated by summing the curvatures of the elevationcurves defining the surface.

The geographic region 102 includes a geographic surface 111 and asubterranean region 104 beneath the surface 111. The subterranean region104 may include various layers of rock and/or other structures. Twoexample layers 106, 108 are shown in FIG. 1A for purposes ofillustration. Generally, the subterranean region 104 may include anynumber of layers and/or other types of geological features, which mayhave any topographical shape, thickness, and/or geometry. For example,the subterranean region 104 may include one or more rock layer havingvarious degrees of porosity, permeability, and/or conductivity, and thesubterranean structures may include faults, fractures, fissures, and/orother types of natural or induced discontinuities. In someimplementations, the subterranean region 104 may contain hydrocarbonresources (e.g., natural gas, oil, coal, etc.), brine water, and/orother types of resources in a reservoir. For example, a subterraneanreservoir may include conventional and/or non-conventional reservoirs.

In some implementations, the geographic region 102 includes all or partof one or more wells (not shown). For example, a well system in thegeographic region 102 may include one or more discovery wells,monitoring wells, injection wells, production wells, and/or other typesof wells. A well system may include a single well bore or multiple wellbores, which may include well bores having vertical, horizontal, slant,curved, and/or other types of geometries. The subterranean region 104may include fluids injected through one or more injection wells and/orinduced fractures generated by the fluid injection. In some instances,one or more sensors or other measurement devices may be installed in awell bore. As such, the measurement subsystem 101 may include one ormore downhole components located in a well bore beneath the surface 111.

The measurement subsystem 101 includes tiltmeter stations 112 in atiltmeter array on the geographic surface 111, GPS receivers 114 in aGPS network on the geographic surface 111, and remote InSAR telemetrysatellites 120. A measurement subsystem 101 may include additionaland/or different components, measurement devices, subsystems, and/orother features. For example, the measurement subsystem 101 may includeother types of measurement systems, such as, for example, aerial InSAR,laser ranging, laser leveling, land surveying, and/or other types ofsystems. The components of the measurement subsystem 101 may be arrangedand/or configured in the manner shown or in a different manner. Forexample, a measurement subsystem may generally include any number oftiltmeter stations 112, any number of GPS receivers 114, and/or othertypes of measurement devices in any type of geometric or schematicarrangement.

Each of the GPS receivers 114 receives signals from GPS satellites andacquires three-dimensional surface coordinate measurements. For example,each of the GPS receivers 114 may acquire a position measurementperiodically, based on predetermined events, based on command signals,and/or based on other criteria. The GPS measurements may indicate asurface elevation at the locations of the GPS receivers 114. Theposition measurements provided by the GPS receivers 114 may includethree independent spatial coordinates, a time coordinate and/or otherinformation. For example, the position measurements may indicate surfaceelevations and surface coordinates (e.g., longitudinal and latitudinalcoordinates) for each of the surface locations of the GPS receivers 114.GPS measurements for two different time points may indicate a measuredtemporal change in elevation at a given surface location.

The InSAR satellites 120 and/or an airborne InSAR system useline-of-sight telemetry to acquire three-dimensional surfacemeasurements. For example, InSAR data from one or more look directionsmay be used to generate three-dimensional surface measurements. TheInSAR satellites 120 transmit electromagnetic signals that interact withthe surface 111. At least a portion of the signals are reflected fromthe surface 111 and received by the satellites 120, and the receivedsignals may be used to generate measurements of surface elevationchanges over time for one or more locations on the surface 111. InSARmeasurements may be acquired periodically, based on predeterminedevents, based on command signals, and/or based on other criteria. TheInSAR measurements may indicate surface elevations changes over time atany surface location for which the InSAR satellites or airborne InSARsystems can acquire a line-of-sight measurement. The measurementsprovided by InSAR systems may include three independent spatialcoordinates, one or more coordinates and/or other information. Forexample, the position measurements may indicate temporal changes insurface elevations and surface coordinates (e.g., longitudinal andlatitudinal coordinates) for various locations on the surface 111.

The tiltmeter stations 112 acquire surface gradient measurements. Thesurface gradient measurements correspond to a temporal change in theslope or “tilt” of the geographic surface 111 at the location of thetiltmeter station 112. The surface gradient can be represented as aunitless quantity that indicates the vertical change of elevation over alateral distance, and/or the surface gradient can be represented as anangular quantity that indicates the surface angle with respect to one ormore reference directions. The surface gradient may be a vector quantitythat includes a directional component. For example, a tiltmeter stationmay measure a surface gradient in the “North” direction, a surfacegradient in the “East” direction, and/or a surface gradient in one ormore other directions. Each of the tiltmeter stations 112 may acquire agradient measurement periodically, based on predetermined events, basedon command signals, and/or based on other criteria. The gradientmeasurements provided by the tiltmeter stations 112 may include surfacelocation coordinates, one or more surface gradient values and anidentification of the gradient directions, one or more time coordinatesand/or other information. For example, the gradient measurements mayindicate surface gradient magnitudes, surface gradient directions, atime period for the measurement, and surface coordinates (e.g.,longitudinal and latitudinal coordinates) for the surface locations ofthe tiltmeter stations 112. Tiltmeter measurements may indicate ameasured temporal change in surface gradient at a given surfacelocation.

In the example shown in FIG. 1A, the tiltmeter stations 112 and the GPSreceivers 114 reside in a region 110 on the geographic surface 111. TheGPS receivers 114 may be interspersed between the tiltmeter stations112. Thus, the GPS receivers 114 may acquire surface elevation data fora first set of surface locations, and the tiltmeter stations 112 mayacquire surface gradient data for a second, different set of surfacelocations. In some implementations, one or more of the GPS receivers 114may reside substantially in the same location with a tiltmeter station112. Thus, a GPS receiver 114 and a tiltmeter station may acquiresurface elevation data and surface gradient data for substantially thesame surface location. In some implementations, one or more of the GPSreceivers 114 may reside substantially apart from the tiltmeter array,for example, outside the region 110. The region 110 can be a region ofany size, shape, or geometry. In some cases, the tiltmeter stations 112and/or the GPS receivers 114 extend over an area ranging from 0.1 to 0.3square miles, and in some cases the region can be smaller (e.g., lessthan 0.1 square mile) or larger (e.g., up to two square miles, orlarger).

Generally, the measurement subsystem 101 may include any number oftiltmeter stations 112 and GPS receivers 114. In some exampleimplementations, the region 110 on the geographic surface 111 mayinclude twenty (20) to two hundred (200) tiltmeter stations 112 and two(2) to five (5) GPS receivers 114. In some implementations, GPSreceivers 114 are not used. In some implementations, the measurementsystem 100 includes a greater number of tiltmeter stations 112 and/or agreater number of GPS receivers 114. In some implementations, thetiltmeter stations 112 and/or the GPS receivers 114 are organizedgenerally in a grid arrangement on the surface 111. In someimplementations, the tiltmeter stations 112 and/or the GPS receivers 114are arranged in a different manner (e.g., radial, linear, random, and/orother types of patterns). The locations of the tiltmeter stations 112and/or the GPS receivers 114 may be selected based on locations ofgeographic features, locations of subterranean features, locations ofother infrastructure and communications equipment, and/or other factors.Data may be collected from some or all of the surface locations at thesame time or at multiple different times.

In some instances, the InSAR satellites 120 can collect elevation dataover substantially all of the geographic surface 111. For example, theInSAR satellites 120 may collect elevation data at surface locationsinside the region 110, and the InSAR satellites 120 may collectelevation data at additional surface locations outside the perimeter ofthe tiltmeter array. As such, the elevation data generated by themeasurement subsystem 101 may include surface elevation information fora large number data points inside and outside the tiltmeter array. Thus,the geodetic data generated by the measurement subsystem 101 may includesurface elevation information without surface gradient information forone or more locations on the surface 111, surface gradient informationwithout surface elevation information for one or more locations on thesurface 111, both surface elevation information and surface gradientinformation for one or more locations on the surface 111, including anycombination.

The measurement subsystem 101 measures temporal changes of thegeographic surface 111. For example, surface elevation informationprovided by the measurement subsystem 101 may include measurements oftemporal changes in surface elevation (e.g., based on measurements bythe GPS receivers 114, the InSAR satellites 120, and/or othermeasurement devices), and surface gradient information provided by themeasurement subsystem 101 may include measurements of temporal changesin surface gradient (e.g., based on measurements by the tiltmeterstations 112, and/or other measurement devices). Temporal changes may bemeasured over any time period. In some instances, each measurementdevice measures temporal changes of the geographic surface 111 overperiods seconds, minutes, hours, days, weeks, months, years, or acombination of these. Temporal changes may be monitored during a timeperiod associated with field activities (e.g., before, during, and/orafter the field activities) to observe the effects of the fieldactivities on the geographic surface 111, if any. A temporal change insurface elevation indicates a relationship between the surface elevationfor two different points in time and does not necessarily indicate adifference in the surface elevation at the two points in time. That isto say, in some instances, the measured temporal change in surfaceelevation can be zero. Similarly, a temporal change in surface gradientindicates a relationship between the surface gradient for two differentpoints in time and does not necessarily indicate a difference in thesurface gradient at the two points in time. That is to say, in someinstances, the measured temporal change in surface gradient can be zero.

The computing subsystem 103 includes data processing systems, devices,and/or components that can store and process information acquired by themeasurement subsystem 101. For example, the computing subsystem 103 mayinclude the example computing system 500 shown in FIG. 5 and/oradditional or different types of systems and devices. The structure andoperation of the example computing system 500 is discussed in moredetail with respect to FIG. 5 below. The computing subsystem 103 mayinclude multiple components in a single location and/or in multipledifferent locations. Some or all components of the computing subsystem103 may be located remotely from the geographic region 102 and/or thecomputing subsystem 103 may include components located at or near themeasurement system 102 in the geographic region 102.

The computing subsystem 103 may include and/or interact withcommunication systems and infrastructure. For example, the computingsubsystem 103 may interact with one or more data networks (e.g., theInternet, a private data network, etc.), telecommunication networks,wired or wireless communication links, and/or other types of interfacesto receive measurement data from the measurement subsystem 101. In someimplementations, some or all of the measurement data and/or relatedinformation may be delivered to the computing subsystem 103 on acomputer-readable medium, such as, for example, a disk, a disk drive, aportable memory device, and/or another type of device.

The computing subsystem 103 may include computer software, applications,modules, codes, functions and/or other types of computer programs thatevaluate the surface data provided by the measurement subsystem 101. Forexample, the computing subsystem 103 may analyze surface data byperforming one or more of the operations in the process 300 shown inFIG. 3. In some example implementations, the computing subsystem 103processes the geodetic data and generates elevation curves alongconnecting lines between data points corresponding to the locations ofthe surface measurements.

FIG. 1B is a schematic diagram 150 showing example geodetic data pointsand connecting lines that may be generated by the surface evaluationsystem 100 shown in FIG. 1A. The squares in the diagram 150 representexample surface elevation data points that correspond to surfacelocations for which surface elevation information is acquired by themeasurement subsystem 101 (e.g., by the GPS receivers 114, by the InSARsatellites 120, etc.). The circles in the diagram 150 represent examplesurface gradient data points that correspond to locations for whichsurface gradient information is acquired by the measurement subsystem101. Three of the surface elevation data points 152 a, 152 b, 152 c andtwo of the surface gradient data points 154 a, 154 b are labeled in thediagram 150. The geodetic data points may correspond to measurementlocations on the geographic surface 111 for which surface gradientinformation and/or surface elevation information has been acquired bythe measurement subsystem 101.

The computing subsystem 103 may receive the geodetic data points asinput data and generate connecting lines between pairs of the datapoints. In some implementations, the connecting lines may correspond toDelaunay lines generated by a Delaunay triangulation of the data points.The connecting lines may be generated by additional and/or differenttechniques. Three example connecting lines 156 a, 156 b, and 156 c areshown in the diagram 150. The computing subsystem 103 may generate asurface elevation curve for each of the connecting lines in the diagram150. The surface elevation curve represents the surface deformation overtime at surface locations along the connecting line. The surfaceelevation curve between two data points may be generated based partiallyon values of surface elevation temporal changes and surface gradienttemporal changes at the two data points.

Because, at least in some instances (as shown, for example, in FIG. 1B),the geodetic data received from the measurement subsystem 101 does notinclude both elevation and gradient information for all of the surfacelocations, the computing subsystem 103 may calculate surface elevationvalues (i.e., temporal changes in surface elevation) for the surfacelocations where the measurement subsystem 101 did not measure a surfaceelevation (e.g., surface gradient data points 154 a, 154 b, etc.),and/or the computing subsystem 103 may calculate surface gradient values(i.e., temporal changes in surface gradient) for the surface locationswhere the measurement subsystem 101 did not measure a surface gradient(e.g., surface elevation data points 156 a, 156 b, 156 c, etc.). Theunmeasured values may be calculated by solving a system of constrainingequations, where the constraining equations are generated based onrelationships between neighboring pairs of the surface locations. Thesystem of constraining equations may include undetermined variables forthe surface elevation or surface gradient at some or all of the surfacelocations. The constraining equations may be designed to generate valuesthat yield the minimum curvature surface based on a given set of datainputs. Example constraining equations are discussed in more detailbelow with regard to FIGS. 2 and 3.

After the elevation and gradient values have been identified for thesurface locations, the computing subsystem 103 may use the elevation andgradient values at each pair of data points to calculate parameters ofthe elevation curve along the connecting line between the data points.For example, the elevation and gradient values may be used to generatecoefficients for the terms of a third order polynomial along eachconnecting line. In some instances, the solution is a set of elevationcurves corresponding to a minimum curvature surface. Examples ofcalculating parameters of the elevation curves between the data pointsare discussed in more detail below with regard to FIGS. 2 and 3. Theelevation curves may be used to calculate elevation, surface curvature,and/or other properties at additional locations on the surface. Forexample, the elevation curves may be interpolated to a Cartesian grid oranother predefined set of points to generate a surface model. Thesurface model represents the deformation of the surface over a timeperiod. Temporal changes represented in the surface may be used toanalyze subterranean resources and structures. For example, movement ofthe Earth's surface may indicate movement of fluids, seismic behaviorand/or other types of events in the layers 106, 108 beneath the Earth'ssurface.

FIG. 2 is a schematic diagram showing example geodetic data points 201,202, 203 and connecting lines 210 a, 210 b, 210 c. The data points 201,202, 203 may correspond to measurements generated by the measurementsubsystem 101 for three surface locations on the geographic surface 111.The connecting lines 210 a, 210 b, 210 c may correspond to Delaunaylines generated by Delaunay triangulation of the surface locations. Forpurposes of explanation, two example scenarios are discussed below withreference to FIG. 2. The techniques described for analyzing the twoexample scenarios may be expanded to additional and/or differentscenarios, which may include substantially any number of surfacelocations and substantially any number of connecting lines. For example,the techniques described for analyzing the two example scenarios may beexpanded to find the minimum curvature surface (i.e., the smoothestsurface) given a number of data points over an area, where some pointshave an input surface gradient measurement, some points have an inputsurface elevation measurement, and some points have input elevation andgradient measurements.

In a first scenario, all three of the geodetic data points 201, 202, 203have a measured gradient (i.e., measurements of temporal changes in thesurface gradients over a given time period). In this scenario, the datapoint 203 can be used as a reference elevation. For a spline fit havinga continuous second derivative, a curve of the formh=ax ³ +bx ² +cx+d  (1)may be assigned to each of the connecting lines 210 a, 210 b, 210 c,where the variable h represents the temporal elevation change along theconnecting line over the given time period and the variable x representsthe distance along the line between x=0 at one of the data points andx=l at the other data point. The measured temporal changes in thesurface gradient (represented by the variables t₁, t₂, and t₃ for thedata points 201, 202, 203 respectively) at the ends of each line relateto the coefficients a, b and c from Equation 1 according to the firstorder derivative of the elevation change h with respect to x:

$\begin{matrix}{\frac{\mathbb{d}h}{\mathbb{d}x} = {{3{ax}^{2}} + {2{bx}} + {c.}}} & (2)\end{matrix}$

For an initial estimate, in the first scenario, there is insufficientinformation to determine all three coefficients, so the lowest orderspline may be used by assuming a=0. In some instances, other assumptionsmay be useful. Evaluating Equation 2 at the boundary values of x yields:

$\begin{matrix}{{{\frac{\mathbb{d}h}{\mathbb{d}x}}_{x = 0} = {c = t_{1}}}{and}} & (3) \\{{\frac{\mathbb{d}h}{\mathbb{d}x}}_{x = l} = {{{2{bl}} + t_{1}} = {t_{2}.}}} & (4)\end{matrix}$From Equations 3 and 4, the elevation curve of Equation 1 may beexpressed

$\begin{matrix}{h = {{\frac{t_{1} - t_{2}}{2l}x^{2}} + {t_{1}x}\; + {d.}}} & (5)\end{matrix}$

Along each of the connecting lines, Equation 5 provides the lowest orderheight calculation. The lowest order height difference for theconnecting line 210 c between data points 201 and 202 is provided by theconstraining equation:

$\begin{matrix}{{{h_{1} - h_{2}} = {\frac{\left( {t_{1} + t_{2}} \right)}{2}l}},} & (6)\end{matrix}$which is equivalent to the average of the gradient measurementsmultiplied by length of the connecting line. Constraining equations canbe generated for the other connecting lines 210 a, 210 b, and theconstraining equations for each connecting line can be expressed inmatrix form, where each column of the matrix represents one of themeasurement points and each row represents one of the connecting lines:

[ - 1 1 0 - 1 0 1 0 - 1 1 ] ⁡ [ h 1 h 2 h 3 ] = ( 1 2 ) ⁡ [ ( t + t ) ⁢ l C( t + t ) ⁢ l ( t + t ) ⁢ l ] . ( 7 )In Equation 7 above, the variables h₁, h₂, and h₃ represent theundetermined values for the temporal changes in the surface elevationsat the data points 201, 202, 203 respectively; the variables l_(A),l_(B), and l_(C) represent the lengths of the connecting lines 210 a,210 b, 210 c respectively; the variable t_(1C) represent the temporalchange in surface gradient at data point 201 along the direction ofconnecting line 201 c; the variable t_(1B) represent the temporal changein surface gradient at data point 201 along the direction of connectingline 210 b; the variable t_(2A) represent the temporal change in surfacegradient at data point 202 along the direction of connecting line 210 a;the variable t_(2C) represent the temporal change in surface gradient atdata point 202 along the direction of connecting line 210 c; thevariable t_(3A) represent the temporal change in surface gradient atdata point 203 along the direction of connecting line 210 a; thevariable t_(3B) represent the temporal change in surface gradient atdata point 203 along the direction of connecting line 210 b. In Equation7, the variables on the right hand side have known values indicated bythe measured geodetic data. Because the left side matrix has rank 2, thematrix cannot be inverted. In this scenario, incorporating a measuredvalue or an assumed reference value of the elevation for the data point203 converts the three-by-three matrix in Equation 7 to a matrix thathas a rank 2. For example, assuming h₃=0 eliminates the third column ofthe left side matrix, and Equation 7 reduces to:

[ - 1 1 - 1 0 0 1 ] ⁡ [ h 1 h 2 ] = ( 1 2 ) ⁡ [ ( t + t ) ⁢ l C ( t + t ) ⁢l ( t + t ) ⁢ l ] ( 8 )

The changes in elevation (h₁ and h₂) at points 201 and 202 may becalculated by solving Equation 8. For example, (h₁ and h₂) may becalculated as

[ h 1 h 2 ] = [ - 1 1 - 1 0 0 1 ] ∖ ( 1 2 ) ⁡ [ ( t + t ) ⁢ l C ( t + t ) ⁢l ( t + t ) ⁢ l ] ( 9 )where the ‘\’ in Equation 9 represents a pseudo-inverse operator thatuses Gaussian elimination to find the least squares solution. Generally,a matrix equation such as Equation 8 may be solved based on invertingthe left-most matrix in the equation. Gaussian elimination andGauss-Jordan elimination are examples of well-known techniques thatgenerate the least squares solution to the matrix equation. Thesetechniques and/or other techniques may be used to solve Equation 8.

Once the elevation changes at the tiltmeter locations are evaluatedusing the least squares solution, the undetermined coefficients (a, b,c, d) in Equation 1 can be calculated, for example, based on Equations2, 3, 4, and 5 above. For connecting line 210 a, Equation 1 becomes

h A = ( ( t 2 ⁢ A + t ) l - 2 ⁢ ( h - h ) l ) ⁢ x 3 + ( 3 ⁢ ( h - h ) l A2 - t + 2 ⁢ t l A ) ⁢ x 2 + t 2 ⁢ A ⁢ x + h 2 ( 10 )Equation 1 may be converted to a similar elevation curve equation forconnecting lines 210 b and 210 c in the first scenario.

Moving to a second scenario, instead of using the assumption h₃=0, thesecond scenario assumes the data point 203 has a measured value for thetemporal change in elevation and an undetermined gradient value. Inother words, in the second scenario, only two of the geodetic datapoints 201, 202 have a measured surface gradient movement, and only oneof the data points 203 has a measured surface elevation movement. In theexample analysis for this second scenario, calculations are similar tothe calculations in the first scenario, except that the gradient is notmeasured at data point 203 and elevation of point 203 is measured. Thetemporal change in the surface gradient at point 203 is represented byundetermined values t_(3E) and t_(3N) corresponding to the surfacegradient in the East and North directions, respectively. In the exampleshown in FIG. 2, only two connecting lines 210 a, 210 b converge atpoint 203, so the gradients of the two line connecting lines 210 a, 210b can be considered independent. In many practical cases, there will bea greater number of lines converging at many of the elevation datapoints. Beginning with the equations:

[ - 1 1 0 - 1 0 1 0 - 1 1 ] ⁡ [ h 1 h 2 h 3 ] = ( 1 2 ) ⁡ [ ( t + t ) ⁢ l C( t + t ) ⁢ l ( t + t ) ⁢ l ] ( 11 )The matrix equation can be extended to account for the measured value ofthe elevation change h₃ and the undetermined values t_(3E) and t_(3N)for the gradient:

[ - 1 1 0 0 - 1 0 - 2 ⁢ sin ⁢ ⁢ θ - 2 ⁢ cos ⁢ ⁢ θ 0 - 1 - 2 ⁢ sin ⁢ ⁢ θ - 2 ⁢ cos ⁢⁢θ ] ⁡ [ h 1 h 2 ] = ( 1 2 ) ⁡ [ ( t + t ) ⁢ l C t 1 ⁢ B ⁢ l B + 2 ⁢ h t ⁢ l A -2 ⁢ h ] ( 12 )

In Equation 12, θ_(3A) represents the angle of the connecting line 210 aat point 203 relative to North, and θ_(3B) represents the angle of theconnecting line 210 b at point 203 relative to North. The left sidematrix has a rank less than 4, so the solution cannot be uniquelycalculated. However, in many situations, the matrix will have enoughlinearly independent rows to allow calculation of a unique least squaresfit. For situations where this is not the case, as here, one option isto further simplify the gradient at the point of measured elevationchange, for example, by assuming t_(3A)=t_(2A) and t_(3B)=t_(1B). Thissimplifies the matrix equation to:

[ - 1 1 - 1 0 0 - 1 ] ⁡ [ h 1 h 2 ] = ( 1 2 ) ⁡ [ ( t + t ) ⁢ l C t 1 ⁢ B ⁢ lB + 2 ⁢ h t ⁢ l A - 2 ⁢ h ] ( 13 )

This simplification acknowledges that the input data does not have aunique solution, so the gradient at the elevation site reverts to thelowest possible curvature, and the remainder of the solution iscalculated from there. Additional and/or different simplifications maybe made in this situation and/or in similar or different situations.

The techniques described with respect to FIG. 2 for these two examplescenarios may be expanded to handle substantially any set of datapoints. For example, the matrices above may be expanded with additionalrows and columns to accommodate tens, hundreds, or thousands of surfacelocations and connecting lines, and any corresponding undeterminedvariables.

FIG. 3 is a flow chart showing an example process 300 for evaluatingsurface data. For example, the process 300 may be used to evaluategeodetic data collected by the measurement subsystem 101 in FIG. 1. Insome implementations, the process 300 may include additional, fewer,and/or different operations performed in the order shown or in adifferent order. Moreover, one or more of the individual operationsand/or subsets of the operations in the process 300 can be performed inisolation and/or in different contexts to achieve a similar or differentresult. In some implementations, one or more of the operations in theprocess 300 may be iterated, repeated, omitted, modified, and/orperformed by multiple sub-operations. Some or all aspects of the process300 may be implemented by data processing apparatus executingcomputer-readable instructions, which may be included in one or moresoftware programs, modules, or applications configured to provide thefunctionality described. While the process 300 is discussed in terms ofdetermining values of temporal changes in elevation and/or temporalchanges in gradient for a time period based on measurements of temporalchanges in elevation and/or temporal changes in gradient for the timeperiod, the process 300 may be used to calculate surface elevationsand/or surface gradients for a given time point based on elevationmeasurements and/or gradient measurements for the given time point.

In some implementations, the process 300 provides a deterministiccalculation of a minimum curvature surface. For example, the process 300may be implemented such that the same output is always generated giventhe same set of input geodetic measurements. A deterministic approachmay be executed faster, for example, compared to some stochasticapproaches that generate a statistically meaningful number of separatesolutions and average them together. While a deterministic approach mayhave certain advantages in some implementations, stochastic techniquesmay also be useful. In some implementations, the process 300 uses aleast squares fit and/or related mathematical techniques to calculatesurface elevations and/or surface gradients. As such, the process 300may inherently provide uncertainty estimates based on how well the leastsquares solution fits the calculated lowest order curvature estimates.

At 302, geodetic data are received. For example, input data can bereceived from a local memory, from a remote device, and/or in adifferent manner. The geodetic data may include surface elevationinformation, surface gradient information, or a combination of surfaceelevation information and surface gradient information at each ofmultiple surface locations. Each of the surface locations may correspondto a measurement location where a measurement was acquired. For example,the geodetic data may include tiltmeter data, InSAR data, GPS data,and/or other types of data. The geodetic data may include measurementsof temporal changes in surface elevations and/or measurements oftemporal changes in surface gradients over a particular time period(e.g., an hour, a week, a month, a year, and/or another time period)and/or over multiple time periods.

At 304, pairs of data points are identified. Neighboring pairs of datapoints may be identified by finding a set of Delaunay lines linking eachof the measurement sites. Delaunay lines may be identified by anytriangulation technique. Delaunay triangulation is an example of aconventional triangulation technique that, given a discrete set ofsurface coordinates, generates connecting lines among neighboring pairsof surface coordinates. The set of connecting lines form triangleshaving vertices at the surface coordinates. The Delaunay triangulationtechnique, in some instances, maximizes the minimum angle of all theangles of the triangles in the triangulation. The pairs of data pointsmay be identified by generating a triangulation that satisfies theso-called “Delaunay condition” that no triangle defines a circumcirclethat encompasses another data point in the triangulation. A Delaunaytriangulation may be generated based on a two-dimensional set ofcoplanar data points. A two-dimensional Delaunay triangulation may begenerated based on data points at different elevations by projecting thedata points onto a plane (or other two-dimensional surface), forexample, based on coordinates of longitude and latitude. In someinstances, a Delaunay triangulation can be generated in three-dimensionsbased on a three-dimensional set of non-coplanar data points. In someinstances, one or more of the Delaunay lines may be selectively ignoredor thrown out, for example, based on the angular density or proximity ofthe Delaunay lines. Additional and/or different techniques may be usedto identify the pairs of data points.

An elevation function may be assigned to each pair of data points. Theelevation function between a pair of data points may represent thetemporal deformation of the surface between the data points during thetime period that the change in elevation and/or change in gradient wasmeasured at one or both of the two data points. For example, theelevation function can be a third order polynomial h=ax³+bx²+cx+d, wherethe variable h represents the temporal change in elevation at each pointalong a path between the data points, the variable x represents thedistance along the path between the data points. The coefficients a, b,c, and d for the elevation function assigned to each pair of data pointsmay initially be undetermined. The third order polynomial describes aspline that may ensure continuous first and second derivatives of thesurface elevation. Additional and/or different types of elevationfunctions may be used.

At 306, constraining equations for surface elevation curves between thepairs of data points are generated. For example, the constrainingequations may be represented in matrix form and/or as another type ofdata object. A set of constraining equations may relate undeterminedvalues for surface gradients (i.e., temporal changes in surfacegradients) and/or undetermined values for surface elevations (i.e.,temporal changes in surface elevations) to measurements of surfaceelevation movement and/or measurements of surface gradient movement inthe received geodetic data. Equations 8 and 12 above provide two examplesets of constraining relationships for three example surface locations.Some or all of the constraining equations can include multipleundetermined values. Equation 6 above is an example of an individualconstraining equation with multiple undetermined elevation values (h₁and h₂) and multiple measured values (t₁ and t₂). A constrainingequation may additionally or alternatively include undetermined gradientvalues and/or measured elevation values in any combination. For aconstraining equation that includes multiple undetermined values, theequation constrains the values of each variable with respect to thevalues of the other variables without independently rendering thevariables determinate. That is to say, a constraining equation mayconstrain without determining unmeasured values. Each of theconstraining equations are typically generated based on a pair of thedata points. As such, the set of constraining equations may include anindividual constraining equation for each pair of data points. In someinstances, constraining equations are generated for fewer than all ofthe pairs.

A constraining equation for a pair of surface locations may relateundetermined values for surface elevations (i.e., undetermined temporalchanges in the surface elevations) at the surface locations to measuredsurface gradient values (i.e., measured temporal changes in the surfacegradients) at the surface locations. In some implementations, for pairsof data points where the surface gradient is known at both ends, theknown surface gradient values may be used to generate the lowest orderchange in elevation along the line connecting the data points. Forexample, for a neighboring pair of surface locations having surfacecoordinates separated by a distance l, where the geodetic data include asurface gradient value t₁ for a first point in the neighboring pair anda surface gradient value t₂ for a second point in the neighboring pair,a constraining equation may constrain an undetermined surface elevationvalue h₁ for the first point and an undetermined surface elevation valueh₂ for the second point by an equation of the form h₂−h₁=½(t₁+t₂)/l.

A constraining equation for a pair of surface locations may relateundetermined values for surface gradients (i.e., undetermined temporalchanges in the surface gradients) at the surface locations to measuredsurface elevation values (i.e., measured temporal changes in the surfaceelevations) at the surface locations. In some implementations, for pairsof data points where the surface elevation is known at both ends, theknown surface elevation values may be used to generate the local surfacegradient that results in a minimum curvature surface. For example, for aneighboring pair of surface locations having surface coordinatesseparated by a distance l, where the geodetic data include a surfaceelevation value h₁ for a first point in the neighboring pair and asurface elevation value h₂ for a second point in the neighboring pair, aconstraining equation may constrain an undetermined surface gradientvalue t₁ for the first point and an undetermined surface gradient valuet₂ for the second point by an equation of the form

${t_{1} + t_{2}} = {\frac{2}{l}{\left( {h_{1} - h_{2}} \right).}}$

A constraining equation for a pair of surface locations may relate anundetermined value for a surface gradient (i.e., an undeterminedtemporal changes in the surface gradient) at a first surface locationand an undetermined value for a surface elevation (i.e., an undeterminedtemporal change in surface elevation) at a second surface location to ameasured surface elevation value (i.e., a measured temporal change inthe surface elevation) at the first surface location and measuredsurface gradient value (i.e., a measured temporal change in the surfacegradient) at the second surface location. In some implementations, forpairs of data points where the surface elevation is known at one end andthe surface gradient is known at the other end, the known surfacegradient value and the known surface elevation value may be used togenerate a constraining equation of the form 2h₂−t₁l=2h₁+t₂l. Forexample, for a neighboring pair of surface locations having surfacecoordinates separated by a distance l, where the geodetic data include asurface elevation value h₁ for a first point in the neighboring pair anda surface gradient value t₂ for a second point in the neighboring pair,a constraining equation may constrain an unknown surface gradient valuet₁ for the first point and an unknown surface elevation value h₂ for thesecond point by an equation of the form 2h₂−t₁l=2h₁+t₂l.

In these and other scenarios, additional and/or different types ofconstraining equation may be used. The gradient values in a constrainingequation may be separated into multiple different terms each havingangular dependencies that represent the directional nature of thegradient. For example, the gradient values may be divided into multiplecomponents each representing the surface gradient along a particularconnecting line based on geometric or trigonometric relationships amongthe connecting lines. An example is provided in Equation 12 above. Anequation for a pair of data points may, in some instances, have only oneundetermined value.

At 308, the constraining equations are solved to identify surfaceelevation values (i.e., temporal changes in the surface elevation)and/or surface gradient values (i.e., temporal changes in the surfacegradient) at the data points where the surface elevation and/or surfacegradient was unknown. For example, the set of constraining equationsdeveloped at 306 may be solved using a least squares method thatgenerates particular surface elevation values for each measurementlocation where the surface elevation was not measured and/or particularsurface gradient values for each measurement location where the surfacegradient was not measured. When the set of constraining equations isrepresented in matrix form, the least squares solution may be generated,for example, by Gaussian elimination, by the Gauss-Jordan technique,and/or other techniques for solving a system of linear equations.

At 310, parameters of surface elevation curves between the pairs of datapoints are determined. For example, with the particular elevationchanges and the particular gradient changes generated based on theconstraining equations, the coefficients (a, b, c, d) of the third orderpolynomial may be uniquely determined for the connecting lines betweensome or all pairs of data points. With particular values for thecoefficients of each elevation curve, the temporal change in elevationat any point along any of the connecting line may be calculated. Theresulting elevation curves may be uniquely determined by the elevationand gradient values determined at 308. In other words, the elevation andgradient values determined at 308 may correspond to a single minimumcurvature surface for the elevation lines between the neighboring pairsof data points. The curvature of a surface can be estimated by summingthe curvatures of each of the triangulation lines between pairs of datapoints.

At 312, the elevation data are interpolated. The coefficients generatedat 310 may be used to interpolate elevation along the connecting lines,for example, to a desired grid and/or other locations. The elevations atthe grid locations may represent a numerical model of the surfacedeformation. At 314, a surface plot is generated. For example, thesurface model generated at 312 may be rendered on display device, aplotter, a printing device, and/or some other medium. FIGS. 4A and 4Bare diagrams showing example surface models that indicate deformation ofan example geographic surface over a time period of one day. Additionaland/or different time periods may be used. FIG. 4A shows an examplesurface model 400 that represents a geographical surface generatedaccording to the example process 300 described above. FIG. 4B showsanother example surface model 450 that represents a geographicalsurface. In the surface model 450 in FIG. 4B, external Delaunaytriangles having an angle below a specified limit have been removed.This allows concave external hull shapes in the arrays, and removesregions where the interpolation of elevation may be invalid.

In some implementations, the surface models generated for each timeperiod can be analyzed to identify movement and/or deformation of thegeographical surface. The movement and/or deformation of thegeographical surface and/or other observed events may be time-correlatedwith field activities. For example, surface deformation may becorrelated with fracturing activities, production activities, drillingactivities, and/or other activities in or near the geographic region. Inother examples, surface deformation may be used to analyze seismicevents and/or movement of geological structures in or near thesubterranean region under the surface. The process 300 may be performedin a relatively short amount of time so that the surface models may bepresented and/or analyzed closer to real time and with greater temporaldetail. As such, increased calculation speed may allow more precisecorrelation of surface deformation to field activities, more preciseanalysis of subterranean events, and/or other advantages.

FIG. 5 is a diagram showing aspects of an example computing system 500.One or more structural or operational aspects of the computing subsystem103 in FIG. 1A may be implemented by the example computing system 500,which may operate in coordination with one or more other computingsystems in additional and/or different locations. In some instances, theexample computing system 500 may perform one or more operations of theexample process 300 shown in FIG. 3. In some instances, the computingsubsystem 500 may generate one or more of the graphical models shown inFIGS. 4A and 4B. The computing subsystem 500 may include additionaland/or different components and may be configured to operate in adifferent manner.

The example computing system 500 includes a processor 512, a memory 510,and input/output controllers 514 communicably coupled by a bus 511. Thememory 510 can include, for example, a random access memory (RAM), astorage device (e.g., a writable read-only memory (ROM) and/or others),a hard disk, and/or another type of storage medium. The computing system500 can be preprogrammed and/or it can be programmed (and reprogrammed)by loading a program from another source (e.g., from a CD-ROM, fromanother computer device through a data network, and/or in anothermanner). The input/output controller 514 is coupled to input/outputdevices (e.g., a monitor 518, a mouse, a keyboard, and/or otherinput/output devices) and to a network 516. The input/output devicesreceive and transmit data in analog or digital form over communicationlinks such as a serial link, wireless link (e.g., infrared, radiofrequency, and/or others), parallel link, and/or another type of link.

The network 516 can include any type of data communication network. Forexample, the network 516 can include a wireless and/or a wired network,a Local Area Network (LAN), a Wide Area Network (WAN), a privatenetwork, a public network (such as the Internet), a WiFi network, anetwork that includes a satellite link, and/or another type of datacommunication network.

The memory 510 can store instructions (e.g., computer code) associatedwith an operating system, computer applications, and/or other resources.The memory 510 can also store application data and data objects that canbe interpreted by one or more applications and/or virtual machinesrunning on the computing system 500. As shown in FIG. 5, the examplememory 510 includes data 530 and programs 540. In some implementations,a memory of a computing device may include some or all of theinformation shown in the example memory 510. The memory 510 may storeadditional information, for example, files and instruction associatedwith an operating system, device drivers, archival data, and/or othertypes of information.

The files and data on the memory 510 include information relating tosurface evaluation such as, for example, geodetic data that includessurface elevation information and/or surface gradient information formultiple surface locations in a geographic region. The informationstored in the memory 510 may include and/or may be derived from datacollected by a remote measurement system, for example, a tiltmeterarray, GPS receivers, an InSAR system, and/or others. In the exampleshown, the memory 510 stores tiltmeter data 532, GPS data 534, InSARdata 536, and curvature data 538. The memory 510 may store additionaland/or different types of information relating to surface elevation.

The tiltmeter data 532, GPS data 534, and InSAR data 536 may includegeodetic data received from, and/or generated based on measurementstaken by, the measurement subsystem 101 in FIG. 1. For example, thetiltmeter data 532 may include surface gradient information generated byone or more tiltmeter stations 112; the GPS data 534 may include surfaceelevation data generated by one or more of the GPS receivers 114; andthe InSAR data 536 may include surface elevation information generatedby the InSAR satellites 120. The geodetic data may include geodetic datafor multiple different geographical regions, as well as geodetic datafor multiple different time periods and/or multiple different surfacelocations in a given geographical region. For example, the tiltmeterdata 532 may include two-dimensional surface coordinates (e.g.,longitude and latitude coordinates) indicating surface locations oftiltmeters that acquired data points included in the tiltmeter data 532,and/or the tiltmeter data 532 may include time data (e.g., relative orabsolute time coordinates) for each data point indicating times when thedata were acquired and/or indicating a time periods over which temporalchange were observed. Similarly, the GPS data 534, the InSAR data 536,and/or other types of geodetic data may include surface coordinate dataand/or time data for each data point. The geodetic data may includeadditional and/or different information. In some instances, the geodeticdata include information on a measurement uncertainty or error bars foreach measurement. In some instances, the geodetic data include a serialnumber, identifier and/or other information on the measurement apparatusthat acquired the data. For example, the geodetic data may includeinformation that identifies the specific tiltmeter station or GPSreceiver that generated the data.

The curvature data 538 may include surface information generated by thesurface analysis program 542 based on geodetic measurements. Forexample, the curvature data 538 may include connecting lines forneighboring surface locations, surface elevation values, and/or surfacegradient values generated by the surface analysis program 542 based onthe tiltmeter data 532, the GPS data 534, the InSAR data 536, and/orother data. The curvature data 538 may include parameters of surfacecurves, for example, along connecting lines between neighboring surfacelocations. In some instances, the curvature data 538 may includeparameters of a polynomial curve (e.g., second order, third order, etc.)between neighboring pairs of data points. The curvature data 538 mayinclude models, diagrams, maps, plots, and/or other types of data thatcan be rendered to generate a visual representation of a geographicalsurface. For example, the curvature data 538 may include one or more ofthe surface models 400, 450 shown in FIGS. 4A and 4B, respectively. Thecurvature data 538 may include information on multiple differentgeographical regions, information on multiple different areas in a givengeographic region, information on multiple different time points, and/orother types of information.

The programs 540 can include software applications, scripts, programs,functions, executables, and/or other modules that are interpreted and/orexecuted by the processor 512. In the example shown, the programs 540includes a surface analysis program 542, which may include softwareapplications, scripts, programs, functions, executables, and/or othermodules that operate alone or in combination as a surface evaluationtool. The surface analysis program 542 may include machine-readableinstructions for performing one or more of the operations shown in FIG.3. The programs 540, including the surface analysis program 542, canobtain input data, such as surface elevation information, surfacegradient information, surface coordinate information, and/or other typesof input data, from the memory 510, from another local source, and/orfrom one or more remote sources (e.g., via the network 516). Theprograms 540, including the surface analysis program 542, can generateoutput data, such as curvature data 538 and/or other types of outputdata, and store the output data in the memory 510, in another localmedium, and/or in one or more remote devices (e.g., by sending theoutput data via the network 516).

The processor 512 can execute instructions, for example, to generateoutput data based on data inputs. For example, the processor 512 can runthe programs 540 by executing and/or interpreting the software, scripts,functions, executables, and/or other modules contained in the programs540. The processor 512 may perform one or more of the operations shownin FIG. 3. The input data received by the processor 512 and/or theoutput data generated by the processor 512 may include any of thetiltmeter data 532, the GPS data 534, the InSAR data 536, the curvaturedata 538, and/or other types of data.

Generally, the surface analysis program 542 can include high-level code,low-level code, source code, object code, machine code, or a combinationof these and/or other types of code. The surface analysis program 542may be written in C, C++, Perl, and/or other types of compiled,interpreted, or executable programming languages. In some exampleimplementations, the surface analysis program 542 may include one ormore functions or files (e.g., a “.m” file) that can be interpretedand/or executed by MATLAB® computational software, available fromMATHWORKS®. Below are some example MATLAB® functions that may beincluded in a “.m” file and/or in multiple related “.m” files. Theexample MATLAB® functions can be executed using MATLAB® version 7.7. Oneor more of the instructions may invoke or otherwise use aMATLAB®-defined function and/or other conventional functions. In someimplementations, one or more of the instructions, functions, and/oralgorithms may be modified, and in some cases additional and/ordifferent instructions, functions, or algorithms may be substituted. Assuch, the following MATLAB® functions provide an example of computerprogram code that may be used to implement aspects of one or more of thetechniques disclosed herein. For example, one or more of the followingMATLAB® functions may be used to perform one or more of the operationsof the process 300 shown in FIG. 3. These and other aspects of thetechniques disclosed herein may additionally or alternatively beimplemented using different types of instructions, different types ofcodes, different types of formulae, different types of algorithms,and/or different types of data objects. The example MATLAB® functionsbelow can be used together. That is to say, some of the instructions inthe example MATLAB® functions below invoke one of the other exampleMATLAB® functions provided below. As such, the inputs and outputs foreach function will be apparent from the context and the accompanyingdescription below.

The following example MATLAB® function may be used to generate a surfacedeformation model based on geodetic data. This example functionidentifies parameters of fitted surface curves based on measurements ofchange in gradient and/or measurements of change in elevation formultiple measured surface locations over a given time period, and theexample function generates an output matrix (“elev”) that includessurface coordinates and values of temporal change in elevation formultiple points on a surface. The output elevation values correspond toa surface of minimum curvature. Additional and/or different techniquesmay be used. The example function may accept the following input dataobjects: “tiltvalues” is a matrix having n rows and four columns, whereeach row includes East and North tiltmeter surface coordinates, an Eastdirection gradient value, and a North direction gradient value;“elevvalues” is a matrix having m rows and three columns, where each rowincludes East and North surface coordinates and an elevation changevalue; “zeropt” is an optional index to a row of the “tiltvalues” matrixto be used as a zero reference elevation; “small_angle_lim” is a valuethat selects Delaunay lines to be removed from the analysis, whereDelaunay triangles with an angle less than small_angle_lim will have thelongest side removed. The inputs “tiltvalues” and “elevvalues” mayinclude measurement data from one or more geodetic measurement systems.

function elev = splinefit(tiltvalues,elevvalues,zsite,small_angle_lim)if nargin == 0  test_spline2;  return; end pointsperline = 10;diagnostics = false; points =organize_inputs(tiltvalues,elevvalues,zsite); triangles =delaunay(points.x,points.y); if diagnostics  openfigure(‘Delaunay’); triplot(triangles,points.x,points.y);  hold on; plot(points.x,points.y,‘or’) end line.ind =[triangles(:,1:2);triangles(:,2:3);[triangles(:,1) triangles(:,3)]];line.ind = unique(sort(line.ind,2),‘rows’); removeline =findsmallangles(triangles,points,small_angle_lim); line.ind =setdiff(line.ind,removeline,‘rows’); if diagnostics  for k =1:size(removeline,1)   x = points.x(removeline(k,:));   y =points.y(removeline(k,:));   plot(x,y,‘r-’)  end end line =get_line_param(points,line); xypts =getxypts(line,points,pointsperline); ideal_delta_h =low_order_spline(line); [A,x] = elev_matrix(points,line); i = ~all(A ==0,2); A = A(i,:); ideal_delta_h = ideal_delta_h(i);[A,x,points,ideal_delta_h] = check_rank(A,x,points,ideal_delta_h,line);solution = A\ideal_delta_h; i = strcmp(x.source,‘elev’);points.elev(x.ind(i)) = solution(i); i = strcmp(x.source,‘etilt’);points.etilt(x.ind(i)) = solution(i); i = strcmp(x.source,‘ntilt’);points.ntilt(x.ind(i)) = solution(i); line.tilt =resolve_tilt(points,line); xypts.elev =fit_curve(points,line,pointsperline); elev = [xypts.x(:) xypts.y(:)xypts.elev(:)]; [temp,nonrepeat] = unique(elev(:,1:2),‘rows’); elev =elev(nonrepeat,:);

The following example MATLAB® function may be used to organize inputgeodetic data for further processing. Additional and/or differenttechniques may be used.

function points = organize_inputs(tiltvalues,elevvalues,zsite) dist_lim= .1; tiltvalues = tiltvalues(~any(isnan(tiltvalues),2),:); elevvalues =elevvalues(~any(isnan(elevvalues),2),:); points.x = tiltvalues(:,1);points.y = tiltvalues(:,2); points.elev = NaN*zeros(size(points.x));points.etilt = tiltvalues(:,3)/1e6; points.ntilt = tiltvalues(:,4)/1e6;points.elev(zsite) = 0; if size(elevvalues,1) < 1  return end for k =1:numel(points.x)  dist =trilength(tiltvalues(k,1:2),elevvalues(:,1:2));  i = find(dist <=dist_lim);  if numel(i) > 1   showwarn(‘Multiple elevation points foundclose to a tilt point’);   [temp,i] = min(dist);  end  if numel(i) == 1  points.elev(k) = elevvalues(i,3);   elevvalues(i,:) = [ ];  end end ifsize(elevvalues,1) < 1  return end points.x =[points.x;elevvalues(:,1)]; points.y = [points.y;elevvalues(:,2)];points.elev = [points.elev;elevvalues(:,3)]; points.etilt =[points.etilt;NaN*zeros(size(elevvalues,1),1)]; points.ntilt =[points.ntilt;NaN*zeros(size(elevvalues,1),1)];

The following example MATLAB® function may be used to remove one or moreconnecting lines based on angles between the connecting lines.Additional and/or different techniques may be used.

function removeline = findsmallangles(triangles,points,anglelimit) xy =[points.x points.y]; a =trilength(xy(triangles(:,1),:),xy(triangles(:,2),:)); b =trilength(xy(triangles(:,2),:),xy(triangles(:,3),:)); c =trilength(xy(triangles(:,3),:),xy(triangles(:,1),:)); anglea = 180/pi *findangle(a,b,c); angleb = 180/pi * findangle(b,a,c); anglec = 180 −(anglea + angleb); smalltri = find(any([anglea angleb anglec] <=anglelimit,2)); [temp,longside] = max([a(smalltri) b(smalltri)c(smalltri)],[ ],2); removeline = zeros(numel(longside),2); for k =1:numel(longside)  if longside(k) == 1,   removeline(k,:) =triangles(smalltri(k),[1 2]);  elseif longside(k) == 2,  removeline(k,:) = triangles(smalltri(k),[2 3]);  elseif longside(k) ==3,   removeline(k,:) = triangles(smalltri(k),[1 3]);  end end removeline= unique(sort(removeline,2),‘rows’);

The following example MATLAB® function may be used to find an anglebased on the law of cosines. Additional and/or different techniques maybe used.

function angle = findangle(a,b,c) angle = acos((c.{circumflex over( )}2 + b.{circumflex over ( )}2 − a.{circumflex over ( )}2)./(2*b.*c));

The following example MATLAB® function may be used to calculate thelength of a connecting line between two surface locations. Additionaland/or different techniques may be used.

function lng = trilength(p1,p2) p1 = p1′; p2 = p2′; lng = sqrt((p1(1,:)−p2(1,:)).{circumflex over ( )}2 + (p1(2,:) − p2(2,:)).{circumflex over( )}2)′;

The following example MATLAB® function may be used to identify agradient, elevation, length and/or orientation for one or more points ona connecting line between two surface locations. Additional and/ordifferent techniques may be used.

function line = get_line_param(points,line) dx =diff(points.x(line.ind),1,2); dy = diff(points.y(line.ind),1,2);line.theta = pi/2 − atan2(dy,dx); line.tilt = resolve_tilt(points,line);line. elev = points.elev(line.ind); p1 = [points.x(line.ind(:,1))points.y(line.ind(:,1))]; p2 = [points.x(line.ind(:,2))points.y(line.ind(:,2))]; line.lng = trilength(p1,p2);

The following example MATLAB® function may be used to identify agradient along a connecting line between two surface locations.Additional and/or different techniques may be used.

function tilt = resolve_tilt(points,line) p1 = line.ind(:,1); p2 =line.ind(:,2); tilt_1 = ...  points.etilt(p1).*sin(line.theta)+points.ntilt(p1).*cos(line.theta);tilt_2 = ...  points.etilt(p2).*sin(line.theta)+points.ntilt(p2).*cos(line.theta);tilt = [tilt_1 tilt_2];

The following example MATLAB® function may be used to generate aconstraining relationship for a pair of neighboring surface locations.Additional and/or different techniques may be used.

function delta_h = low_order_spline(line) delta_h =NaN*ones(size(line.lng)); i = all(isnan(line.elev),2); delta_h(i) =−mean(line.tilt(i,:),2).*line.lng(i); i = ~any(isnan(line.elev),2);delta_h(i) = −diff(line.elev(i,:),1,2); i = isnan(line.elev(:,1)) & ...  ~isnan(line.elev(:,2)) & isnan(line.tilt(:,2)); delta_h(i) =−line.tilt(i,1).*line.lng(i)/2 − line.elev(i,2); i =isnan(line.elev(:,1)) & ...   ~isnan(line.elev(:,2)) &~isnan(line.tilt(:,2)); delta_h(i) =−mean(line.tilt(i,:),2).*line.lng(i) − line.elev(i,2); i =~isnan(line.elev(:,1)) & ...   isnan(line.tilt(:,1)) &isnan(line.elev(:,2)); delta_h(i) = −line.tilt(i,2).*line.lng(i)/2 +line.elev(i,1); i = ~isnan(line.elev(:,1)) & ...  ~isnan(line.tilt(:,1)) & isnan(line.elev(:,2)); delta_h(i) =−mean(line.tilt(i,:),2).*line.lng(i);

The following example MATLAB® function may be used to generate a set ofconstraining relationships based on geodetic data. Additional and/ordifferent techniques may be used.

function [A,x] = elev_matrix(points,line) A =zeros(size(line.ind,1),numel(points.x)); x.source(1:size(A,2)) ={‘elev’}; x.ind(1:size(A,2)) = 1:numel(points.elev); for k =1:numel(line.lng)  A(k,line.ind(k,1)) = −1;  A(k,line.ind(k,2)) = 1; endi = isnan(points.elev); A = A(:,i); x.source = x.source(i); x.ind =x.ind(i); i = find(~isnan(points.elev) & isnan(points.etilt)); orig_col= size(A,2); % Active column of A; A = [A zeros(size(A,1),2*numel(i))];x.source = [x.source cell(1,2*numel(i))]; x.ind = [x.indzeros(1,2*numel(i))]; for k = 1:numel(i)  rows = any(line.ind ==i(k),2);  col = orig_col + 1 + 2*(k−1);  x.source(col:col+1) = {‘etilt’‘ntilt’};  x.ind(col:col+1) = i(k);  A(rows,col) =line.lng(rows).*sin(line.theta(rows))/2;  A(rows,col+1) =line.lng(rows).*cos(line.theta(rows))/2; end i = ~all(A == 0,1); A =A(:,i); x.source = x.source(i); x.ind = x.ind(i);

The following example MATLAB® function may be used to determine if amatrix can be inverted and/or to interpolate an undetermined surfacegradient at one or more surface locations, for example, to increase therank of the matrix. Additional and/or different techniques may be used.

function [A,x,points,ideal_delta_h] = ...     check_rank(A,x,points,ideal_delta_h,line) if rank(A) >= size(A,2) return end reduce_count = 0; while rank(A) < size(A,2)  reduce_count =reduce_count + 1;  col = find(strcmp(x.source,‘etilt’),1);  i =x.ind(col);  conn_lines = any(line.ind == i,2);  conn_pts =line.ind(conn_lines,:);  conn_pts = conn_pts(conn_pts ~= i);  theta =line.theta(conn_lines);  Ap = [sin(theta) cos(theta)];  line_tilt =line.tilt(conn_lines,:);  line_lng = line.lng(conn_lines);  Bp =NaN*zeros(sum(conn_lines),1);  for k = 1:numel(Bp)   ifany(~isnan(line_tilt(k,:)))    Bp(k) =line_tilt(k,~isnan(line_tilt(k,:)));   elseif~isnan(points.elev(conn_pts(k)))    Bp(k) = (points.elev(i) −points.elev(conn_pts(k)))/line_lng(k);   end  end  Ap =Ap(~isnan(Bp),:);  Bp = Bp(~isnan(Bp));  if rank(Ap) == size(Ap,2)  tilt = Ap\Bp;  else   showwarn(‘Could not resolve tilt’)  end points.etilt(i) = tilt(1);  points.ntilt(i) = tilt(2);  ideal_delta_h =ideal_delta_h − A(:,col+[0 1])*tilt(:);  keepindex =setdiff(1:size(A,2),col+[0 1]);  A = A(:,keepindex);  x.source =x.source(keepindex);  x.ind = x.ind(keepindex);  line.tilt =resolve_tilt(points,line); end if reduce_count == 1  str =[‘Interpolated tilt at ’ num2str(reduce_count) ‘ location.’]; else  str= [‘Interpolated tilt at ’ num2str(reduce_count) ‘ locations.’]; enddisp([‘Inversion matrix was rank deficient.’ str])

The following example MATLAB® function may be used to calculate pointson an elevation curve between two surface locations. Additional and/ordifferent techniques may be used.

function point_elev = fit_curve(points,line,pointsperline) n =numel(line.lng); end_elev = points.elev(line.ind); alpha = zeros(n,4);tilt = −line.tilt; alpha(:,1) = (tilt(:,1) +tilt(:,2))./line.lng.{circumflex over ( )}2 − ...     2*diff(end_elev,1,2)./line.lng.{circumflex over ( )}3; alpha(:,2) =3*diff(end_elev,1,2)./line.lng.{circumflex over ( )}2 − ...     (2*tilt(:,1) + tilt(:,2))./line.lng; alpha(:,3) = tilt(:,1);alpha(:,4) = end_elev(:,1); point_elev = zeros(n,pointsperline); for k =1:n  xval = linspace(0,line.lng(k),pointsperline);  point_elev(k,:) =polyval3(alpha(k,:),xval); end

The following example MATLAB® function may be used to calculate pointson a third order polynomial elevation curve between two surfacelocations. Additional and/or different techniques may be used.

function out = polyval3(coeff,value) out = coeff(1).*(value.{circumflexover ( )}3) + coeff(2).*(value.{circumflex over ( )}2) + ...    coeff(3).*value + coeff(4);

The following example MATLAB® function may be used to calculate surfacecoordinates along a connecting line between two surface locations.Additional and/or different techniques may be used.

function xypts = getxypts(line,points,pointsperline) numlines =size(line.ind,1); xypts.x = zeros(numlines,pointsperline); xypts.y =zeros(numlines,pointsperline); xypts.elev =zeros(numlines,pointsperline); for k = 1:numlines  xypts.x(k,:) =linspace(points.x(line.ind(k,1)), ...       points.x(line.ind(k,2)),pointsperline);  xypts.y(k,:) =linspace(points.y(line.ind(k,1)), ...       points.y(line.ind(k,2)),pointsperline); end

Some aspects of the subject matter and the operations described in thisspecification can be implemented in digital electronic circuitry, or incomputer software, firmware, or hardware, including the structuresdisclosed in this specification and their structural equivalents, or incombinations of one or more of them. Some aspects of the subject matterdescribed in this specification can be implemented as one or morecomputer programs, i.e., one or more modules of computer programinstructions, encoded on computer storage medium for execution by, or tocontrol the operation of, data processing apparatus. Alternatively or inaddition, the program instructions can be encoded on anartificially-generated propagated signal, e.g., a machine-generatedelectrical, optical, or electromagnetic signal, that is generated toencode information for transmission to suitable receiver apparatus forexecution by a data processing apparatus. A computer storage medium canbe, or be included in, a computer-readable storage device, acomputer-readable storage substrate, a random or serial access memoryarray or device, or a combination of one or more of them. Moreover,while a computer storage medium is not a propagated signal, a computerstorage medium can be a source or destination of computer programinstructions encoded in an artificially-generated propagated signal. Thecomputer storage medium can also be, or be included in, one or moreseparate physical components or media (e.g., multiple CDs, disks, orother storage devices).

Operations described in this specification can be implemented asoperations performed by a data processing apparatus on data stored onone or more computer-readable storage devices or received from othersources.

The term “data processing apparatus” encompasses all kinds of apparatus,devices, and machines for processing data, including by way of example aprogrammable processor, a computer, a system on a chip, or multipleones, or combinations, of the foregoing. The apparatus can includespecial purpose logic circuitry, e.g., an FPGA (field programmable gatearray) or an ASIC (application-specific integrated circuit). Theapparatus can also include, in addition to hardware, code that createsan execution environment for the computer program in question, e.g.,code that constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, a cross-platform runtimeenvironment, a virtual machine, or a combination of one or more of them.The apparatus and execution environment can realize various differentcomputing model infrastructures, such as web services, distributedcomputing and grid computing infrastructures.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, object, orother unit suitable for use in a computing environment. A computerprogram may, but need not, correspond to a file in a file system. Aprogram can be stored in a portion of a file that holds other programsor data (e.g., one or more scripts stored in a markup languagedocument), in a single file dedicated to the program in question, or inmultiple coordinated files (e.g., files that store one or more modules,sub-programs, or portions of code). A computer program can be deployedto be executed on one computer or on multiple computers that are locatedat one site or distributed across multiple sites and interconnected by acommunication network.

Aspects of the processes and logic flows described in this specificationcan be performed by one or more programmable processors executing one ormore computer programs to perform actions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application-specific integrated circuit).

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andany one or more processors of any kind of digital computer. Generally, aprocessor will receive instructions and data from a read-only memory ora random access memory or both. The essential elements of a computer area processor for performing actions in accordance with instructions andone or more memory devices for storing instructions and data. Generally,a computer will also include, or be operatively coupled to receive datafrom or transfer data to, or both, one or more mass storage devices forstoring data, e.g., magnetic, magneto-optical disks, or optical disks.However, a computer need not have such devices. Moreover, a computer canbe embedded in another device, e.g., a Global Positioning System (GPS)receiver, or a portable storage device (e.g., a universal serial bus(USB) flash drive), and other types of devices. Devices suitable forstoring computer program instructions and data include all forms ofnon-volatile memory, media and memory devices, including by way ofexample semiconductor memory devices, e.g., EPROM, EEPROM, and flashmemory devices; magnetic disks, e.g., internal hard disks or removabledisks; magneto-optical disks; and CD-ROM and DVD-ROM disks. Theprocessor and the memory can be supplemented by, or incorporated in,special purpose logic circuitry.

To provide for interaction with a user, aspects of the subject matterdescribed in this specification can be implemented on a computer havinga display device, e.g., a CRT (cathode ray tube) or LCD (liquid crystaldisplay) monitor, for displaying information to the user and a keyboardand a pointing device, e.g., a mouse or a trackball, by which the usercan provide input to the computer. Other kinds of devices can be used toprovide for interaction with a user as well; for example, feedbackprovided to the user can be any form of sensory feedback, e.g., visualfeedback, auditory feedback, or tactile feedback; and input from theuser can be received in any form, including acoustic, speech, or tactileinput. In addition, a computer can interact with a user by sendingdocuments to and receiving documents from a device that is used by theuser; for example, by sending web pages to a web browser on a user'sclient device in response to requests received from the web browser.

Some aspects of the subject matter described in this specification canbe implemented in a computing system that includes a back-end component,e.g., as a data server, or that includes a middleware component, e.g.,an application server, or that includes a front-end component, e.g., aclient computer having a graphical user interface or a Web browserthrough which a user can interact with an implementation of the subjectmatter described in this specification, or any combination of one ormore such back-end, middleware, or front-end components. The componentsof the system can be interconnected by any form or medium of digitaldata communication, e.g., a communication network. Examples ofcommunication networks include a local area network (“LAN”) and a widearea network (“WAN”), an inter-network (e.g., the Internet), andpeer-to-peer networks (e.g., ad hoc peer-to-peer networks).

The computing system can include clients and servers. A client andserver are generally remote from each other and typically interactthrough a communication network. The relationship of client and serverarises by virtue of computer programs running on the respectivecomputers and having a client-server relationship to each other. In someembodiments, a server transmits data (e.g., an HTML page) to a clientdevice (e.g., for purposes of displaying data to and receiving userinput from a user interacting with the client device). Data generated atthe client device (e.g., a result of the user interaction) can bereceived from the client device at the server.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of any whatmay be claimed, but rather as descriptions of features specific toparticular implementations. Certain features that are described in thisspecification in the context of separate implementations can also beimplemented in combination in a single implementation. Conversely,various features that are described in the context of a singleimplementation can also be implemented in multiple implementationsseparately or in any suitable subcombination. Moreover, althoughfeatures may be described above as acting in certain combinations andeven initially claimed as such, one or more features from a claimedcombination can in some cases be excised from the combination, and theclaimed combination may be directed to a subcombination or variation ofa subcombination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various systemcomponents in the implementations described above should not beunderstood as requiring such separation in all implementations, and itshould be understood that the described components and systems cangenerally be integrated together in a single product or packaged intomultiple products.

In the present disclosure, “each” refers to each of multiple items oroperations in a group, and may include a subset of the items oroperations in the group and/or all of the items or operations in thegroup. In the present disclosure, the term “based on” indicates that anitem or operation is based at least in part on one or more other itemsor operations—and may be based exclusively, partially, primarily,secondarily, directly, or indirectly on the one or more other items oroperations.

A number of embodiments of the invention have been described.Nevertheless, it will be understood that various modifications may bemade without departing from the spirit and scope of the invention.Accordingly, other embodiments are within the scope of the followingclaims.

What is claimed is:
 1. A system comprising: a database that storesgeodetic data associated with a geographic region, the geodetic dataincluding surface gradient information for at least a subset of aplurality of surface locations in the geographic region, wherein thegeodetic data comprise surface elevation information for a first subsetof the surface locations and surface gradient information for a secondsubset of the surface locations; and a processor operable to performoperations including: generating a set of constraining relationshipsbased on the geodetic data, the set of constraining relationshipsrelates undetermined values for temporal changes in surface gradients atthe first subset of surface locations and undetermined values fortemporal changes in surface elevations at the second subset of surfacelocations to the surface elevation information and the surface gradientinformation included in the geodetic data, a plurality of theconstraining relationships each include multiple undetermined values formultiple surface elevations; and identifying particular values fortemporal changes in surface elevations at each surface location in thesubset comprises identifying particular values for temporal changes insurface gradients for each of the first subset of locations andparticular values for temporal changes in surface elevation each of thesecond subset of surface locations based on determining a solution tothe set of constraining relationships.
 2. The system of claim 1, furthercomprising a measurement subsystem that acquires the geodetic data for aplurality of measurement locations, each of the plurality of surfacelocations corresponding to one of the measurement locations.
 3. Thesystem of claim 2, the measurement subsystem comprising at least one ofa Global Positioning System (GPS) or an Interferometric SyntheticAperture Radar (InSAR) system that generates the surface elevationinformation, the surface elevation information including measurements oftemporal changes in surface elevations at the first subset of surfacelocations.
 4. The system of claim 2, the measurement subsystemcomprising a tiltmeter array that generates the surface gradientinformation, the surface gradient information including measurements oftemporal changes in surface gradients at the first subset of surfacelocations.
 5. The system of claim 1, the operations further comprisinggenerating a model of geographic surface deformation in the geographicregion based on the received geodetic data and the particular values. 6.The system of claim 1, wherein the geodetic data include measurements oftemporal changes in surface gradients over a plurality of time periods,wherein the processor is operable to perform operations furthercomprising: generating a set of constraining relationships for each ofthe plurality of time periods; and identifying particular values fortemporal changes in surface elevations for the subset of surfacelocations for each time period based on determining a solution to theset of constraining relationships for each time period.
 7. The system ofclaim 6, the operations further including generating models ofgeographic surface deformation for each time period; and correlatingtemporal changes of the geographic surface with field activitiesassociated with the geographic region.
 8. The system of claim 7, theoperations further including analyzing subterranean geological featuresbased on the temporal changes of the geographic surface.